Two finite sets of points in the plane are called mutually avoiding if any
straight line passing through two points of anyone of these two sets does not
intersect the convex hull of the other set. For any integer n, we construct a
set of n points in general position in the plane which contains no pair of
mutually avoiding sets of size more than O (n). The given bound is tight up to
a constant factor, since Aronov et al. [AEGKKPS] showed a polynomial time
algorithm for finding two mutually avoiding sets of size (n) in any set of n
points in general position in the plane.